The Green-Schwarz model of the superstring is in contrast to the NSR-string model (the original spinning string), which has manifest worldsheetsupersymmetry but no manifest spacetime supersymmetry. It is a non-trivial theorem that the spectrum of the NSR-string enjoys spacetime supersymmetry (after GSO projection) and may hence be identified with perturbative excitations of a supergravity background. The construction of the Green-Schwarz functional was motivated by the desire to find an equivalent alternative formulation in which spacetime supersymmetry is manifest (see e.g. Schwarz 16, slides 24-25).

The first step in this implication (identifying the spinning string as the superstring) is fairly straightforward (in fact this is how the concept of supersymmetry was discovered in “the west”, in the first place), but the second step (that the superstring excitations necessarily are quanta of a spacetimesupergravity theory) appears as a miracle from the point of view of the Neveu-Schwarz-Ramond superstring. It comes out this way by non-trivial computation, but is not manifest in the theory.

where (xa,θα)(x^a, \theta^\alpha) are the canonical coordinates on ℝd−1,1|N\mathbb{R}^{d-1,1\vert \mathbf{N}}, with the odd-graded elements {θα}\{\theta^\alpha\} spanning the given real Spin(d-1,1)-representationN\mathbf{N} with Clifford algebra generators {Γa}\{\Gamma^a\}.

the key phenomenon of supersymmetry (that two fermions pair to a bosons) means that ℝd−1,1|N\mathbb{R}^{d-1,1\vert \mathbf{N}} is slightly non-abelian, reflected by the fact that the super-vielbein is not closed

is a non-trivial superLie algebra cocycle on ℝd−1,1|N\mathbb{R}^{d-1,1\vert \mathbf{N}}, in that dμF1=0\mathbf{d}\mu_{F1} = 0 and so that there is no left invariant differential formbb with db=μF1\mathbf{d}b = \mu_{F1} (beware here the left-invariance condition: there are of course non-left-invariant potentials for μF1\mu_{F1}, and in fact these are exactly the possible Lagrangian densities for the WZW action functional SWZWS_{WZW}).

In order to get rid of the restriction to some chartU⊂XU \subset X one needs to add global data. The need for this is at least mentioned briefly in (Witten 86, p. 261 (17 of 20)), but seems to have otherwise been ignored in the physics literature. The general solution is to promote the local potentials BB to the connection B^\hat B on a super gerbe (FSS 13). This is a choice of higher prequantization

This form of the Green-Schwarz action functional for the string has evident generalization to other p-branes. Whenever there is a Spin(d-1,1)-invariant(p+2)(p+2)-cocycle μp+2\mu_{p+2} on ℝd−1,1|N\mathbb{R}^{d-1,1\vert \mathbf{N}}, then one may ask for a higher gerbe (higher prequantum line bundle) C^\hat C with curvatureμp+2X\mu^X_{p+2} and consider the analogous functional.

The triples (d,N,p)(d,\mathbf{N},p) (spacetime dimension, number of supersymmetries, dimension of brane) such that

The graphics on the left is from (Duff 87). The diagonal lines indicate double dimensional reduction, taking a (p+1)(p+1)-brane in (d+1)(d+1) dimensions to a pp-brane in dd-dimensions.

For instance for (d=11,N=32,p=2)(d = 11, \; \mathbf{N} = \mathbf{32}, \; p = 2) one finds a cocycle, and the corresponding GS-action functional is that of the fundamental M2-brane.

This was a striking confluence of brane physics and classification of superLie algebra cohomology. But just as striking as the matching, was what it lacked to match: the D-branes and the M5-brane (d=11d = 11, p=5p = 5) are lacking from the old brane scan. Incidentally, these lacking branes are precisely those branes on which the branes that do appear on the brane scan may end, equivalently those branes that have higher gauge fields on their worldvolume (tensor multiplets).

An action functional for the M5-brane analogous to a Green-Schwarz action functional was found in (BLNPST 97, APPS 97). It is again the sum of a kinetic term and a WZW-like term, but the WZW-like term does not come from a cocycle on a (super-)group.

Each item in this bouquet denotes a super L-infinity algebra and each arrow denotes an L-infinity extension classified by a cocycle which encodes the GS-WZW term of the brane named by the domain of the arrow. Moreover, arrows pass exactly from one brane species to the brane species that may end on the former.

This phenomenon is indeed a consequence of the fundamental Green-Schwarz branes:

Consider a 1/2-BPS state solution of type II supergravity or 11-dimensional supergravity, respectively. These solutions locally happen to have the same classification as the Green-Schwarz branes. Hence we may consider a configuration ϕ:Σ→X\phi \colon \Sigma \to X of the corresponding fundamental pp-brane which embeds Σ\Sigma into the asymptotic AdS boundary of the given 1/2 BPS spacetime XX. Then it turns out that restricting the Green-Schwarz action functional to small fluctuations around this configuration, and applying a diffeomorphismgauge fixing, then the resulting action functional is that of a supersymmetricconformal field theory on Σ\Sigma as in the AdS-CFT dictionary:

Here the “⇙\swArrow” filling the triangles is the non-trivial gauge transformation by which the WZW term (as any WZW term) is preserved under the symmetries (instead of being fixed identically). It is the information in this transformations which makes the currents form an extension of the symmetries.

Here this yields the famous brane charge extensions of the super-isometry super Lie algebra of the schematic form

Notice that there are “microscopic degrees of freedom” of the theory encoded by the choice of generalized cohomology theory here, generalizing the extra degrees of freedom in the choice of a WZW-term already mentioned above. In general for EE a cohomology theory and E⟶E⊗ℚE \longrightarrow E \otimes \mathbb{Q} its Chern character map (for instance from topological K-theory to ordinary cohomology in every second degree), then a choice of genuine charges is the extra information encoded in a lift

Above we saw that the naive cocycles of the D-branes and of the M5-brane are not defined on the actual spacetime, but on some “extended” spacetime, which is really a smooth super infinity-groupoid extension of spacetime. Hence we should ask if these cocycles descend to the actual super-spacetime while picking up some twists.

One may prove that:

the F1/Dpp-brane GS-WZW cocycles descend to 10d type II superspacetime to form a single cocycle in rational twisted K-theory, just as the traditional lore reqires (Fiorenza-Sati-Schreiber 16);

may be read as saying that ee is torsion-free except for that term. Notice that this term is the only one that appears when the differential is applied to “Lorentz scalars”, hence to object in CE(𝔰𝔦𝔰𝔬)CE(\mathfrak{siso}) which have “all indices contracted”. (See also at torsion constraints in supergravity.)

This relation is what govers all of the exceptional super Lie algebra cocycles that appear as WZW terms for the Green-Schwarz action below: for some combinations of (D,p)(D,p) a Fierz identity implies that the term

Dimensions – the brane scan

The Green-Schwarz action functional of a pp-brane propagating on an dd-dimensional target spacetimes makes sense only for special combinations of (p,d)(p,d), for which there are suitanble super Lie algebra cocycles on the super translation Lie algebra (see above).

The corresponding table has been called the brane scan in the literature, now often called the “old brane scan”, since it has meanwhile been further completed (see below). In (Duff 87) the “old brane scan” is displayed as follows.

where the first summand is the super-Lie algebra cocycle that classifies the supergravity Lie 3-algebra and the second is the field strength of the supergravity C-field proper (hence a purely bosonic differential form). In the second line we have rewritten this more manifestly in terms of the super-vielbein(EA)=(Ea,Eα)=(Ea,Ψα)(E^A) = (E^a, E^\alpha) = (E^a, \Psi^\alpha), this way the expression is directly analogous to that of definite 3-forms in the theory of G2-manifolds (see this example for details).

Therefore it is natural to consider the perturbation of the Green-Schwarz sigma-models around their asymptotic embeddings into AdS spaces, hence effectively the perturbation theory of the degrees of freedom at those naked singularity at which the corresponding black brane sits.

AdS5AdS_5

The super 3-cocycle for the Green-Schwarz superstring on the super anti de Sitter spacetimeAdS5×S5AdS_5 \times S^5 (i.e. on SU(2,2|5)Spin(4,1)×SO(5)\frac{SU(2,2 \vert 5)}{Spin(4,1)\times SO(5)}) is originally due to

These authors amplify the role of closed (p+2)(p+2)-forms in super pp-brane backgrounds (p. 3) and clearly state the consistency conditions for the M2-brane in a curved backround in terms of the Bianchi identities on p. 7-8, amounting to the statment that the 4-form field strength has to be the pullback of the cocycle ψ¯∧ea∧eb∧Γabψ\overline{\psi}\wedge e^a \wedge e^b \wedge \Gamma^{a b} \psi plus the supergravity C-fieldcurvature and has to be closed.

That the heterotic supergravity equations of motion are sufficient for the 3-form super field strength HH to be closed was first argued in

A more comprehensive result arguing that the heterotic supergravity equations of motion of the background are not just sufficient but also necessary for (and hence equivalent to) the heterotic GS-string on that background being consistent was then claimed in